The exact solution of the generalized Black-Scholes model for pricing derivative securities, for arbitrary payoff at a predetermined terminal liquidation date, is presented. This covers all derivative securities except those that may have, and allow, optimal liquidation before the terminal date. Exact solutions for both time-independent and time-dependent parameters are derived. Use of the solution is illustrated for a few known model solutions. The impact of time-dependent parameters on implied volatility calculations is discussed.

Abstract The exact solution of the generalized Black-Scholes model for pricing derivative securities, for arbitrary payoﬀ at a predetermined terminal liquidation date, is presented. This covers all derivative securities except those that may have, and allow, optimal liquidation before the terminal date. Exact solutions for both time-independent and time-dependent parameters are derived. Use of the solution is illustrated for a few known model solutions. The impact of time-dependent parameters on implied volatility calculations is discussed.

1

I. Introduction A basic equation used for pricing derivative securities is a generalization of the BlackScholes equation [7],

(

∂f ∂t

) + γ (θ, t)θ
θ

(

∂f ∂θ

)

1 + σ 2 (θ, t)θ2 2 t

(

∂2f ∂θ2

) = r(t)f
t

(1)

where γ (θ, t) = µ(θ, t) − λ(θ, t)σ (θ, t), µ(θ, t) is the expected growth rate of the underlying variable θ, λ(θ, t) is the market price of risk of θ, σ (θ, t) is the volatility of θ, and r(t) is the risk-free rate of return. As made explicit by the notation, the general case permits the parameters to be dependent on the time and the underlying variable, with the exception of the risk-free rate of return, which in most cases should be independent of the underlying variable. Risk neutral valuation assumes that γ (θ, t) is independent of risk preferences, that is, the risk preferences contained in µ(θ, t) are eliminated by the subtraction of λ(θ, t)σ (θ, t). It should be remembered that, in Eq. (1) above, θ may represent anything, whether a ﬁnancial security or not. For the case that θ represents a stock paying a continuous dividend, Eq. (1) becomes [7]

Since Eqs.(2) and (3) can be obtained from Eq. (1) with the appropriate interpretation of the variables, the notation of Eq. (1) will be used in the following analysis. The derivative security may be liquidated, or has optimal liquidation value, only at the terminal liquidation time. This is a generalized terminology that is meant to include diverse securities with their accompanying vocabulary. For instance, if the derivative security is a forward contract, then the terminal liquidation time is the delivery date: liquidation or delivery takes place only on the liquidation or delivery date. In the case of options, the terminal liquidation time is the expiration date and liquidation means exercise of the option: optimal liquidation value at the terminal liquidation time means there is no motive for early exercise. The solution of Eq. (1) must satisfy two boundary conditions, typically one at the terminal liquidation time T for arbitrary values of θ and one for a speciﬁed value of θ, say θ = 0, for arbitrary times t. In other words, f (θ, T ) and f (0, t) are assumed to be known functions. Note that, as long as (∂f /∂θ)t and (∂ 2 f /∂θ2 )t are ﬁnite as θ → 0, then, as θ approaches zero (∂f /∂t)θ → r(t)f and ∫T
t

f (0, t) = e

−

dt′ r (t′ )

f (0, T )

(4)

Since f (0, T ) can be obtained from f (θ, T ), the payoﬀ at liquidation f (θ, T ) will completely determine the solution for any pricing problem for which liquidation cannot occur before the terminal liquidation time and for which the minimum value of the underlying variable θ is zero. The solution to these equations, subject to the boundary conditions, usually proceeds after a particular payoﬀ function f (θ, T ) has been speciﬁed. This paper, however, will present a wide class of solutions, subject to the above stated boundary conditions and for 3

arbitrary payoﬀ: those for time-independent and θ-independent parameters (γ (θ, t) = γ , σ 2 (θ, t) = σ 2 , and r(t) = r) and those for time-dependent but θ-independent parameters (γ (θ, t) = γ (t), σ 2 (θ, t) = σ 2 (t), and r(t)). Another problem of interest, the θ-dependent case with or without time-dependence, which extends the constant elasticity of variance model, is not addressed in this paper. The advantage of the arbitrary payoﬀ solution is two-fold: (1) it presents a uniﬁed approach to derivative securities and (2) it permits the development of prototype derivative products by simply designing payoﬀ functions with desired properties at liquidation. The solutions given here are related to the Green’s function approach [12], implicitly providing the Green’s function for the terminal date liquidation and suboptimal early liquidation problems. The solution for the general non-stochastic time-dependence problem provides a boundary condition, in the non-stochastic limit, for models of stochastic behavior of γ (t), σ 2 (t), and r(t). A possible extension to include stochastic behavior of the parameters will only be brieﬂy discussed and will not be pursued in detail in this paper. In Section II, the exact solution to Eq. (1) subject to boundary condition Eq. (4) is derived for arbitrary payoﬀ and constant γ (t), σ 2 (t), and r(t). A brief discussion of the implications of early liquidation is also presented. In Section III, the exact solution to Eq. (1) for arbitrary payoﬀ, subject to boundary condition Eq. (4), is derived for the general time-dependent case. Stochastic behavior for the time-dependent parameters is brieﬂy discussed. In Section IV, applications of the exact solution in Section III are presented for some known model derivatives, reproducing and extending known results to illustrate the utility of the approach. In Section V, implications of the general time-dependent solution for implied volatility calculations are outlined. Section VI contains concluding remarks.

4

II. Time-independent Parameters The simplest approach to deriving an analytical solution to the basic pricing equation is to assume the parameters in Eq. (1) are constants. That is, γ (t) = γ , σ (t) = σ , and r(t) = r are assumed to be independent of time. Following the original approach of Black and Scholes [4], a change of variables is deﬁned by

f (θ, t) = e−r(T −t) g (u(θ, t), v (t))

(5)

u(θ, t) =

2 1 (γ − σ 2 ) ln(θ/θ0 ) + v (t) 2 σ 2

(6)

v (t) =

2 1 (γ − σ 2 )2 (T − t) 2 σ 2 ) =
u

(7)

(

∂g ∂v

(

∂2g ∂u2

) (8)
v

where θ0 , r, γ , and σ are assumed to be constants, independent of time. While this is the transformation originally used by Black and Scholes, it may be of interest to note that θ0 in Eq. (6) is arbitrary and need not be identiﬁed with a strike price in an option. It may also be of interest to note that the somewhat less restrictive transformation equations, ( ) 1 2 u(θ, t) = A ln(θ/θ0 ) − (r − σ )t + u(θ0 , 0) 2

(6a)

v (t) =

σ2 2 A (T − t) 2 5

(7a)

with A, u(θ0 , 0), and θ0 arbitrary constants, also generates Eq. (8). In addition to the constants A, u(θ0 , 0), and θ0 , both Eqs. (7) and (7a) each contain another constant that has been determined by the desired condition that the transformed time variable obey v (T ) = 0, establishing an initial value problem in the transformed coordinates. The boundary condition at t = T translates to f (θ, T ) = g (u(θ, T ), 0). Note that −∞ < u < ∞ since 0 < θ < ∞ while 0 ≤ v ≤ v (0). Optimal early liquidation would limit both the range of v (t) and, more importantly, θ and u(θ, t). Early liquidation or exercise for a put requires θp ≤ θ < ∞ and up ≤ u < ∞ where θp is the value of θ for which early exercise is optimal and up is the corresponding value for the transformation variable u; early liquidation or exercise for a call requires 0 ≤ θ ≤ θc and −∞ < u ≤ uc where θc is the value of θ for which early exercise is optimal and uc is the corresponding value for the transformation variable u. It may be noted in passing that, for the case of a callable bond, the domain of the bond price is also limited from above: the maximum price of the bond, the sum of the coupons plus the face value, limits the domain of θ so that 0 ≤ θ ≤ θb where θb is the maximum value of the bond price. This condition suggests that the call premium for a callable bond may be related to an American call with optimal early exercise for θ = θb . This will be true only if the maximum bond price is greater than the strike price of the option: if the maximum bond price is less than the strike price, the option will never be exercised and should have a zero value for all values of time t. The exact solution to the diﬀusion equation, Eq. (8), subject to the translated boundary condition is [1,5]

(10) √ √ 1 α = (γ − σ 2 )(T − t) + 2ησ T − t 2 This is the exact solution to the derivative security pricing equation, restricted by a lack of motive for early liquidation, and subject to the boundary conditions that f (θ, T ) is a known function and f (0, t) = exp(−r(T − t))f (0, T ). It is of the Green’s function solution form [12] with the Green’s function for the diﬀusion equation reduced to the simple exponential by means of integration variable transformations. Evaluation of Eq. (10) for a designed payoﬀ gives the valuation of a prototype product and investigation of the valuation can then help decide whether the prototype has desirable features that may be marketable and, consequently, whether more eﬀort should be concentrated on developing the prototype into a new ﬁnancial investment vehicle. III. General Time-dependent Case Hull gives a verbal description of the solution to the Black-Scholes equation for timedependent parameters but does not present the solution nor does he discuss the implications of the time-dependent nature of the parameters [7]. Hull and White [8], in treating stochastic volatility, present a solution for the Black-Scholes equation for a European call with time-dependent volatility: the solution appears to involve the Black-Scholes call solution with the constant variance replaced by an integral of σ 2 (t) over all time from t = 0 to the liquidation or expiration time T . As will be shown below, this is not quite correct. It should be pointed out that solutions can be obtained for any deterministic, modeled, or presumed known behavior for the time-dependence of the parameters. This may include a 7

restricted form of stochastic behavior for interest rates and volatilities by means of modeling the time-dependence with pseudo-random functions of time. While such solutions are not practical as pricing tools unless the time-dependent behavior of the parameters can be predicted or modeled with suﬃcient accuracy, these solutions are particularly useful tools in the analysis of historical data since they can explicitly handle time-dependent data in a self-consistent fashion. The solution of the diﬀerential equations presented in Section I can be generalized by allowing γ (t), r(t), and σ (t) to be arbitrary, but integrable, functions of time t. A more general coordinate transform that still leads to the diﬀusion equation is ∫T
t

f (θ, t) = e [

−

dt′ r (t′ )

g (u(θ, t), v (t)) ]

(11)

∫
t

T

u(θ, t) = A ln(θ/θ0 ) +

1 dt′ (γ (t′ ) − σ 2 (t′ )) 2

(12)

1 v (t) = A2 2 ( )

∫
t

T

dt′ σ 2 (t′ )

(13)

∂g ∂v

( =

u

∂2g ∂u2

) (14)
v

where A and θ0 are arbitrary constants. This transformation reduces to the original BlackScholes transformation (Eqs. (6) and (7)) for constant parameters, γ (t′ ) = γ , r(t′ ) = r, and σ (t′ ) = σ independent of time, and by choosing the constant A to be (2/σ 2 )(γ − σ 2 /2)2 . As long as γ (t), r(t), and σ (t) have a ﬁnite number of discontinuities and have integrable, if any, singularities, then the transform exists and solution to the problem can be obtained as in Section II. 8

Note that r ˆ(t), γ ˆ (t), and σ ˆ 2 (t) are the average values of the respective parameters over the interval [t, T ] [8]. It is a simple matter to verify that Eq. (15) reduces to Eq. (10) for the case of time-independent parameters. Eq. (15) is simply Eq. (10) with r replaced by r ˆ(t), γ replaced by γ ˆ (t), and σ 2 replaced by σ ˆ 2 (t). It is curious to note that this is precisely the solution that would be obtained if the time-dependent parameters γ (t), σ 2 (t), and r(t) in the diﬀerential equation were replaced by γ ˆ (t), σ ˆ 2 (t), and r ˆ(t) but treated as constants. Eq. (15) points out one of the complexities, mentioned above, of pricing derivative securities with the generalized Black-Scholes model: using time-dependent parameters is a strong impetus for beginning at time t = 0, making use of known initial values of the timedependent parameters and projecting parameter values forward in time, then integrating 9

the pricing equation forward in time; this procedure, however, requires the parameter values for all values of t from t to T to generate the time-integrated parameters γ ˆ (t), σ ˆ 2 (t), and r ˆ(t). On the other hand, calculation of the value of the derivative security at time t = 0, using the known payoﬀ value at time t = T and requiring parameter values for all t between t and T , is equally strong impetus for beginning at time T and then integrating backward in time; this procedure, however, requires accurate knowledge of the basic parameters γ (t), σ 2 (t), and r(t) at the liquidation time T in order to project values from the future into the past. Nevertheless, as a product design tool, Eq. (15) is more generally applicable and more powerful than Eq. (10) although, like Eq. (10), it simply requires design of a payoﬀ function with desired properties at liquidation. The general time-dependence of Eq. (15) permits greater testing of a new product’s response to timedependent changes in the parameters aﬀecting the valuation of the new product. In other words, Eq. (15) permits more depth of exploration in advanced and engineering design phases of the development of a new derivative product.

Perhaps the most important aspect of Eq. (15) relevant to a stochastic treatment of the parameters γ (t), σ 2 (t), and r(t) is that of a boundary condition: any solution for stochastic parameters must, in the limit of non-stochastic time-dependence, reproduce Eq. (15). However, one problematic aspect of the Garman equation [7] should be noted. The generalized Black-Scholes equation, Eq. (1), can be recovered from the Garman equation by noting that for a non-stochastic parameter the partial derivative with respect to the parameter holding time t and the underlying variable θ constant is zero since the parameter can not be varied if both the time and the underlying variable are held constant in the non-stochastic case. However, if the Garman equation is solved, the only apparent way to recover the non-stochastic solution is by some limiting process. This would appear to 10

require the limit as the variance of the parameter goes to zero to eliminate the stochastic component, and yet there is no reason to expect a zero variance for a time-dependent parameter. Further, it would appear that the limit of zero mean growth rate of the parameter may also be required and, in addition, the order of limits may also be important. In a study of the stochastic volatility problem, Hull and White [8] obtained a formal solution of the Garman equation of the form ∫ f (θ, t, [V (t)]) =
0

∞

dV0 fBS (θ, t, [V0 ])h(V (t), V0 , S, t)

(A)

where fBS (θ, t, [V0 ]) is the solution to the Black-Scholes equation, Eq. (3), for variance V0 and V (t) = σ 2 (t) is the stochastic variance. The function h(V (t), V0 , S, t) is interpreted as the conditional distribution for variance V0 given variance V (t). However, to derive this result, it is suﬃcient to assume a product form for the solution with a separation constant V0 : this requires the function h(V (t), V0 , S, t) to obey a rather complex equation derivable from the Garman equation,

[( fBS

∂h ∂t

) + rS
S,V

(

∂h ∂S

)
t,V

1 + V S2 2

(

∂2h ∂S 2

) +µ ˆV
t,V

(

∂h ∂V

)
t,S

1 + ξ2V 2 2

(

∂2h ∂V 2

)
t,S

]

(B ) 1 + (V − V0 )hS 2 2 ( ∂ 2 fBS ∂S 2 ) +VS
t 2

(

∂fBS ∂S

) (
t

∂h ∂S

) =0
t,V

where fBS = fBS (S, t, [V0 ]) and h = h(V, V0 , S, t). It should be remembered that the Garman equation results from assuming the model stochastic process dV = µV dt + ξV dz for the variance, where µ is the mean rate of growth of V , ξ is the volatility of V , and dz is the Wiener process. The variable µ ˆ is µ − λξ , where λ is market price of risk of V . This is 11

clearly not an advance over the original Garman equation unless a number of assumptions are made concerning the form or behavior of the function h(V, V0 , S, t) [8]. IV. Applications Eq. (15), with the appropriate interpretation of variables, can handle all derivative securities on an underlying variable with an unlimited range of values and which can or should not be liquidated until the terminal liquidation time. For instance, by setting γ (t) = r(t) − rf (t) where rf (t) is the risk-free interest rate in foreign currency and interpreting θ as the spot exchange rate, it provides the Black-Scholes solution for currency options [7]; or by choosing γ (t) = 0 and interpreting θ as the futures price, Eq. (15) gives the solution for the Black model for futures options [3,7]. To illustrate the explicit use of the solution Eq. (15), four simple cases will be presented: (a) the European call, (b) the European put, (c) the American call on a nondividend paying stock, and (d) the forward contract. All the solutions will generalize the known results for constant parameters (γ , σ 2 , and r) to solutions for time-dependent parameters. a. European call. The payoﬀ for the European call is [2,4,7]

where P (x) is the standard normal distribution. Eq. (21) gives the exact solution for a European call on an arbitrary underlying quantity θ. It is easy to see that it also reproduces the known solution for the case of a non-dividend paying stock [2,4,7] (γ (t) = r(t), γ ˆ (t) = r ˆ(t)) as well as the solution for the case of a stock paying a continuous dividend [7] at rate ∫T ′ ′ ˆ ( t) = r ˆ(t) − q ˆ(t)), both generalized q (t) with q ˆ(t) = (T 1 −t) t dt q (t ), (γ (t) = r (t) − q (t), γ to include time-dependent parameters. b. European put. The payoﬀ for a European put is [2,4,7,9]

fEp (θeα , T ) = Max(X − θeα , 0)

(22)

where Max(x, y ) = the maximum of x and y , the subscript Ep means European put and X is the strike price in the put option. For the solution Eq. (15), this restricts the range of integration by 13

d2 η < −√ 2 where d2 is given above in Eq.(20). Upon substitution in Eq. (15), this yields

Eq. (24) is the exact solution for a European put on an abritrary underlying variable θ. It reduces to the known solutions for a European put on a non-dividend paying stock [2,4,7,9] (γ (t) = r(t)) and on a stock paying a continuous dividend [7] at rate q (t) (γ (t) = r(t)−q (t)), both generalized to include time-dependent parameters. c. American call on non-dividend paying stock. This case satisﬁes the same boundary condition as a European call for θ tending to zero. If early exercise is suboptimal, then the boundary condition at the expiration time must also be satisﬁed and then the solution for the American call is identical to that for the European call. On the other hand, if early exercise is optimal, then the value of the European call must intersect the intrinsic value gc (θ) = θ − X at some value of the stock price, say Kc (t). However, the intrinsic value of the call option, for t < T , is always less than the value of the European call [11],

gc (θ) ≤ fEc (θ, t)

(25)

The equality holds only for t = T . This means that Kc (t) exists only for t = T and that Kc (T ) = X . In other words, there is no motive for early exercise and the call is exercised only at expiration for stock prices greater than the exercise price. Consequently, the boundary condition at expiration for the European call applies to the American call as well which then implies that the American call is identical to the European call [7,11]: 14

ˆ(t)(T −t) fAc (θ, t) = θP (d1 ) − Xer P (d2 )

(26)

The inequality Eq.(25) supplies the proof that, in this case, the boundary conditions satisﬁed by the European and American calls are identical. Uniqueness then guarantees that there is only one solution of the diﬀerential equation subject to the stated boundary conditions. In contrast to the American call option, the American put option on a non-dividend paying stock does have early exercise possibilities. If early exercise is optimal, then the intrinsic value of the put, gp (θ) = X − θ, must intersect the value of the European put at some value of stock price, say Kp (t). For stock prices below Kp (t) the intrinsic value of the put is greater than the European put. In particular,

gp (0) ≥ fEp (0, t)

(27)

which proves that a non-zero Kp (t) does exist. In other words, for θ ≤ Kp (t), the value of the European put is less than the intrinsic value which is the motive for early exercise. The American put must satisfy a boundary condition, together with the high contact condition, not at θ = 0 but at some value θ = KA (t) [9]: fAp (KA (t), t) = X − KA (t) and (∂fAp (θ, t)/∂θ)t |θ=KA (t) = −1, where KA (t) is diﬀerent from Kp (t). One of these conditions is a boundary condition on the solution for the option value f (θ, t) while the other condition is a self-consistent equation that determines the optimal exercise price KA (t). These conditions can not be accomodated in the above solution since the coordinate transformation assumes 0 ≤ θ < ∞, that is, the domain of the stock price is assumed to be semi-inﬁnite; to provide for early exercise of the put, the domain of the stock price must be restricted, KA (t) ≤ θ < ∞. 15

d. Forward Contract. The payoﬀ function for a forward contract [7,10], not an option on a forward contract, is

ff (θeα , T ) = θeα − X

(28)

where X is the contracted delivery price. The integrations are then unrestricted in range and result in

γ (t)−r ˆ(t))(T −t) ˆ(t)(T −t) ff (θ, t) = θe(ˆ − Xe−r

(29)

This is the general time-dependent relation between the forward contract price f and the spot price θ. It reduces to the known relation for a contract on a security with a known dividend yield q (t) (γ (t) = r(t) − q (t)) [7,10], but generalizing it to the time-dependent case.

V. Implied Volatility While use of Eq. (15) may not greatly aid the pricing of derivative securities in a direct manner since this requires knowledge of the time-dependent parameters for all relevant times, Eq. (15) should be of value in calculating implied volatilities [2,7,13]. In particular, it should be noted that for time-dependent parameters, it is not suﬃcient to use the Black-Scholes solution for constant parameters and then use diﬀerent values of the parameters for diﬀerent times: integrals of the parameters, Eqs. (16) - (18), are needed to correctly account for the time-dependent behavior. That is, r ˆ(t) and γ ˆ (t) must be input into the equation, not r(t) and γ (t): for the special case of strict time-independence of the parameters, of course, there is no distinction. Furthermore, it is σ ˆ 2 (t) and not σ 2 (t) that 16

is directly obtained in the implied volatility calculation: further calculation is required to extract the variance, and consequently the volatility, from σ ˆ 2 (t),

σ 2 (t) = σ ˆ 2 (t) − (T − t)

dσ ˆ 2 (t) dt

(30)

Only in the special, but highly unlikely event, that σ ˆ 2 (t) is not dependent on time t, is the implied variance directly obtained in the traditional calculation. Consequently, it should be no surprise that the implied volatilities obtained from the standard Black-Scholes analysis are not found to be the observed volatilities [13].

VI. Conclusion This paper presents exact solutions for pricing derivative securities in the generalized Black-Scholes model for arbitrary payoﬀ but without motive for early liquidation. Exact solutions are derived for both time-independent (Eq. (10)) and time-dependent (Eq. (15)) parameter cases. The relation of the time-dependent parameter solution to the stochastic parameter problem is brieﬂy outlined. Known solutions for model problems are obtained from the general solution in a straight forward way. The problematic results of implied volatility calculations using derivative prices is illumined and shown not be problematic at all: the traditional calculations do not actually obtain the volatility, but the time integrals of the variance (Eq. (18)). In addition, the traditional approach inputs, at best, the basic time-dependent parameters (γ (t) and r(t)) in the implied volatility calculation rather than the average of the parameters over the interval [t, T ] (Eqs. (16) and (17)).